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F) = z(f), z(0) = zt l so = zocC and u zo = Ie-C. 23)),vzeget ' t ztz,t)= 4(zc-t). C onsidertheCauchy problem ztt+ tztz,fl' tzz= 5(z,f), ziER,t> 0, ' tzlt r,0)= /(z), z iER. 31)to computethesolutionsalong the characteristics. j.. 33)with initialdata ' tztz,0)= 4(z) 'ztlz,0)= ' t ? )(z). ) - ' /ztz,f). By the chain rule,w e get ô1 dp tzz = vqdz + vvldz = vq+ vvl ' uzz = ' &c + 2170 + vrlr). J#v? ) - /($)+ . ). ûaresupposedto begiven f-unctions. 39)wehave tztz,t) = /(z+ t)+ g(z- t) ' and uztz,f)--/'(z4-f)- g'lz- f).

F) Usetheproceduredescribedabovetoestimatetherateofconvergence forthesequence given by zs= 1//log(/z)I. Try to explain the difficultiesyou encounter,and notethedangersof blindly applying the procedure. 9nrhenraylorseriesisusefulhere. Try to estim ate the order ofconvergenceforthe following series by applying thetechnique developed above. 2 é? Anton (11. ,ckhave to be zero,thevectorsare referred to asa linearly independentset. PROBLEM S (a) Definethevectors - 0 1 2). 32 Setting the Scene (b) Show thatthevectors Singular and N onsingular M atrices.

The vectorsdefined by the rows of-4 form a linearly dependent set. zsuch thatytz = 0. zsuch thatthelinearsystem ytz = b has no solution. Obviously,therank ofa nonsingularn,x n, m atrix is zz. PROBLEMS(CONTINUED) (d) Let 1 2 - 1 ytz= 1 12 13 12 13 14 13 14 15 4 7 5 8 6 9 . - 12 14 1 12 12 34 36 192 - 180 - , ytlzl= 51 and where51= (1,2,1)T'and 52= (-1,2,-4)T'. (f) Show thattherank of-41,-% ,and ytais3,3,and2respectively. (g) Show thatifad# 5c,then and the associated norm isdefined by II zlI- (z,z)1/2.

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